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number, entity describing the magnitude or position of a mathematical object or extensions of these concepts.

The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of objects if their members can be matched in a one-to-one correspondence. Ordinal numbers refer to position relative to an ordering, as first, second, third, etc. The finite cardinal and ordinal numbers are called the natural numbers and are represented by the symbols 1, 2, 3, 4, etc. Both types can be generalized to infinite collections, but in this case an essential distinction occurs that requires a different notation for the two types (see transfinite number).

The Integers and Rational Numbers

To the natural numbers one adjoins their negatives and zero to form the integers. The ratios a/b of the integers, where a and b are integers and b ≠ 0, constitute the rational numbers; the integers are those rational numbers for which b = 1. The rational numbers may also be represented by repeating decimals; e.g., 1/2 = 0.5000 … , 2/3 = 0.6666 … , 2/7 = 0.285714285714 … (see decimal system).

The Real Numbers

The real numbers are those representable by an infinite decimal expansion, which may be repeating or nonrepeating; they are in a one-to-one correspondence with the points on a straight line and are sometimes referred to as the continuum. Real numbers that have a nonrepeating decimal expansion are called irrational, i.e., they cannot be represented by any ratio of integers. The Greeks knew of the existence of irrational numbers through geometry; e.g., 2 is the length of the diagonal of a unit square. The proof that 2 is unable to be represented by such a ratio was the first proof of the existence of irrational numbers, and it caused tremendous upheaval in the mathematical thinking of that time.

The Complex Numbers

Numbers of the form z = x + yi, where x and y are real and i = −1, such as 8 + 7i (or 8 + 7−1), are called complex numbers; x is called the real part of z and yi the imaginary part. The real numbers are thus complex numbers with y = 0; e.g., the real number 4 can be expressed as the complex number 4 + 0i. The complex numbers are in a one-to-one correspondence with the points of a plane, with one axis defining the real parts of the numbers and one axis defining the imaginary parts. Mathematicians have extended this concept even further, as in quaternions.

The Algebraic and Transcendental Numbers

A real or complex number z is called algebraic if it is the root of a polynomial equation zⁿ + a_{n − 1}z^{n − 1} + … + a₁z + a₀ = 0, where the coefficients a₀, a₁, … a_{n − 1} are all rational; if z cannot be a root of such an equation, it is said to be transcendental. The number 2 is algebraic because it is a root of the equation z² + 2 = 0; similarly, i, a root of z² + 1 = 0, is also algebraic. However, F. Lindemann showed (1882) that π is transcendental, and using this fact he proved the impossibility of "squaring the circle" by straight edge and compass alone (see geometric problems of antiquity). The number e has also been found to be transcendental, although it still remains unknown whether e + π is transcendental.

Bibliography

See G. Ifrah, The Universal History of Numbers (1999).

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number

Basic element of mathematics used for counting, measuring, solving equations, and comparing quantities. They fall into several categories. The counting numbers are the familiar 1, 2, 3 . . . ; whole numbers are the counting numbers and zero; integers are the whole numbers and the negative counting numbers; and the rational numbers are all possible quotients formed by integers, including fractions. These numbers can be symbolically represented by terminating or repeating decimals. Irrational numbers cannot be represented by fractions of integers or repeating decimals and must be represented by special symbols such as √2, e, and π. Together, the rational and irrational numbers constitute the real numbers, which form an algebraic field (see field theory), as do the complex numbers. While the counting numbers and rational numbers come about as the means of counting, calculating, and measuring, the others arose as means of solving equations. See also transcendental number.

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number

1. a concept of quantity that is or can be derived from a single unit, the sum of a collection of units, or zero. Every number occupies a unique position in a sequence, enabling it to be used in counting. It can be assigned to one or more sets that can be arranged in a hierarchical classification: every number is a complex number; a complex number is either an imaginary number or a real number, and the latter can be a rational number or an irrational number; a rational number is either an integer or a fraction, while an irrational number can be a transcendental number or an algebraic number

2. the symbol used to represent a number; numeral

3. 

a. a self-contained part of an opera or other musical score, esp one for the stage